Heretofore, the variable gain circuit of this type was fabricated in many cases based on an approximation, termed a bilinear transform relation, represented by the following approximation (1):
                                          ⅇ                          2              ⁢                                                          ⁢              x                                ≈                                    1              +              x                                      1              -              x                                      =                  1          +                      2            ⁢                                                  ⁢            x                    +                      2            ⁢                          x              2                                +                      2            ⁢                          x              3                                +                      …            ⁢                                                  ⁢                          (                                                -                  1                                <                x                <                1                            )                                                          (        1        )            
With the bilinear transform relation, shown by the above relation (1), attention is to be directed to the fact that it is not ex but e2x that the relation approximates.
On the other hand, an exponential function is represented by
                              ⅇ          x                =                  1          +          x          +                                    x              2                        2                    +                                    x              3                        6                    +          …          +                                    x              n                                      n              !                                +          …                                    (        2        )            and also is represented by
                              ⅇ          x                =                              1            +                          tanh              ⁡                              (                                  x                  /                  2                                )                                                          1            -                          tanh              ⁡                              (                                  x                  /                  2                                )                                                                        (        3        )            as an identity that exploits a hyperbolic function (tan h(x)).
This identity also appears in a set of formulas and may be found with ease from the well-known definition of the hyperbolic function (tan h(x)) by the following equation:
                              tanh          ⁡                      (            x            )                          =                                            ⅇ              x                        -                          ⅇ                              -                x                                                                        ⅇ              x                        +                          ⅇ                              -                x                                                                        (        4        )            
That is, the following equation:
                              ⅇ                      2            ⁢                                                  ⁢            x                          =                              1            +                          tanh              ⁡                              (                x                )                                                          1            -                          tanh              ⁡                              (                x                )                                                                        (        5        )            may be arrived at with ease.
However, as far as the present inventor is aware, the above equation (3) or (5) has not been disclosed in treatises or Patent Documents directed to the variable gain circuit with the exponentially varying gain, with the exception of Patent Document 1 (JP Patent Kokai Publication No. JP-P2003-179447A) by the same inventor as the present inventor.
Comparing the equations (2) and (1), it may readily be surmised that the approximation error will become larger.
For example, if the equation (5) is approximated to
                              tanh          ⁡                      (            x            )                          ≈                  x          +                                                    x                3                            3                        ⁢                                                  ⁢                          (                                                                  x                                                  ⁢                                  <<                  1                                            )                                                          (        6        )            we obtain
                              ⅇ                      2            ⁢                                                  ⁢            x                          =                                            1              +                              tanh                ⁡                                  (                  x                  )                                                                    1              -                              tanh                ⁡                                  (                  x                  )                                                              ≈                                    1              +              x              +                                                x                  3                                3                                                    1              -              x              -                                                x                  3                                3                                                                        (        7        )            
The relationship between the original function e2x and its approximations (1) and (7) is shown in FIG. 1, from which it is apparent that the approximation error of the bilinear transform relation, represented by the approximation (1), is of a significantly large value. It is also noted that the bilinear transform relation, represented by the approximation (1), is to be used as an approximation for the exponential function e2x in a range of x of approximately −0.5<x<0.5.
Thus, if a circuit is fabricated on the basis of the approximation termed the bilinear transform approximation (1), the approximation error becomes larger and cannot be decreased except by narrowing down the range of the input voltage.    [Patent Document 1] JP Patent Kokai Publication No. JP-P2003-179447A